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In the usual description of the instanton of nonabelian gauge theory in $D=4$ spacetime, we always (or just usually?) choose the $D=4$ Euclidean spacetime see for example https://en.wikipedia.org/wiki/BPST_instanton.

In the interpretation of $D=4$ nonabelian gauge theory, we often freely Wick rotate the Euclidean spacetime to Minkowski spacetime.

My question --- do we really have a physical and mathematical rigorous description of instanton of nonabelian gauge theory in Minkowski spacetime? Or does the instanton only make sense in the Euclidean spacetime?

ann marie cœur
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    Related: https://physics.stackexchange.com/q/323456 – Nihar Karve Mar 07 '21 at 03:19
  • Thanks, I focus on $D=4$ of nonabelian gauge theory, the other post is other dimensions. – ann marie cœur Mar 07 '21 at 14:23
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    It is not clear to me what kind of "description" you are looking for here. As Qmechanic explains at length in his answer to the question Nihar linked, it is meaningless/irrelevant to "Wick rotate" individual solutions to the equations of motion. But individual solutions to the equations of motion are irrelevant in QFT to begin with! We're interested in stuff like the n-point functions, which you get by analytically continuing the n-point functions from the Wick-rotated Euclidean theory. So what do you think is missing? – ACuriousMind Mar 09 '21 at 16:19
  • We can adjust the issue to the Wick-rotated correlation functions of the instanton solutions, from Euclidean to Minkowski. Does this make sense to you? (thanks for comments!) – ann marie cœur Mar 10 '21 at 14:44

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